There is a Western Rite Vicariate of the Antiochain jurisdiction which use a modified Rite of St. Gregory or sometimes called the Rite of St. Tikhon, named after the Russian Patriarch who modified it and excised the portions that were not in accordance with Orthodox teaching and practice.
There are only a handful of western rite churches within the Antiochian jurisdiction. Most are Eastern Rite. The only other jurisdiction that has western rite parishes is the ROCOR (Russian Orthodox Church Outside of Russia) and these are mainly in Europe. The other jurisdictions such as the Greek, Serbian, etc. do not have western rite parishes and the reason is that they believe that the Eastern Rite is the exclusive rite of the Eastern Orthodox jurisdictions.
You should also be informed that the use of the western rite is not to be regarded as a "bridge to the Western Church." It isn't. The use of the Rite of St. Gregory or St. Tikhon is an Orthodox Rite that has its origins way before the schism. Granted, in the West more was added to it that was clearly heretical, but that was excised by St. Tikhon. THe Western Rite is Orthodox. It is not supposed to be a bridge so that the parishioners will come over to the Eastern Rite, though some often do. Western Rite parishes often struggle to survive. The one here in Omaha, unfortunately, is. Here is more information authored by Bishop BASIL of the Antiochian Diocese of Wichita and Mid-America: http://www.westernorthodox.com/basil.htm